Physics 220 Class 30

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Physics 220 Class 30

• Today you will learn
• how accelerating charges affect circuits in
  significance ways
• about induced electric fields
• about induced magnetic fields and displacement
  current
• the meaning of Faradays Law
• the meaning of Maxwells Term

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Reminders

• Chapter Test 9 due today
• HW 10 due Thursday
• Chapter Test 10 due Friday

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A Summary of the Important Points

• Acceleration fields drop off as 1/r rather than
  1/r2.
• The electric field, the magnetic field, and
  are all mutually perpendicular.
• The vector points in the direction of
  .
• B is smaller than E by a factor of c in SI units.

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Radiation Fields Qualitatively

• If charges are moving slowly, the basic equations
  for the acceleration fields of point charges are

is the vector from the source to the field
point, as in Chapter 8.
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What Were Going to Do

• To find quantitative results, we would have to
  slice sources into small regions and integrate
  over source distributions as we did in Chapter 8.
  (Except we have to be very careful about the time
  threads are emitted!)
• Instead, we are going to qualitatively describe
  the fields that are produced. For this, were
  mostly interested in directions

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The General Plan

• Find the part of the charge or current
  distribution that contributes most strongly to
  the fields at a point P.
• Find the direction of the electric and/or the
  magnetic field at P.
• Make flagrant generalizations.

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Example 1 A Wire with Increasing Current

• In a long, cylindrical wire, current travels to
  the right. Current is increasing in time.
• When current increases, positive charge carriers
  experience an acceleration in the direction of
  the current.

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Current and Velocity

• Assume the density of conduction electrons, ?, is
  known.
• Let T be the time it takes an electron to travel
  a distance L.

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Current, Velocity, and Acceleration

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Finding the Electric Field

• Choose a field point P.

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Finding the Electric Field

• Find the part of the wire that contributes most
  to the fields.

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Finding the Electric Field

• Draw the vectors and

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Finding the Electric Field

• Find

P
(Into the screen)
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Finding the Electric Field

• Find

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Finding the Magnetic Field

• Find

(out of the screen)
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Finding

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Induced Current

• This can cause current to flow in an adjacent
  wire.

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Induced Current

• or an adjacent loop.

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Example 2 A Loop with Increasing Current

• A loop works much the same as a straight wire

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A Loop with Increasing Current

• If the current is increasing

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A Loop with Increasing Current

• If the current is increasing

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Induced Current in a Loop

• If the current is increasing

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A Loop with Increasing Current

• The electric field we form in here is a new type
  of electric field that forms a loop. It resembles
  the magnetic field in this way.

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The Curl of the Magnetic Field

• Magnetic fields are caused by a current. At a
  point in space where looping magnetic fields are
  formed, we found that the curl was proportional
  to the current density

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The Curl of the Electric Field

• At a point in space where the electric field
  loops are formed, the only thing present is the
  magnetic field of the wire.
• We cant think of the magnetic field itself as
  the source of looping electric fields, as
  constant fields dont produce such electric
  fields.
• The source is not the magnetic field, but the
  change in the magnetic field

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Faradays Law of Induction

• This is Faradays Law of Induction in
  Differential Form. It means that at any point in
  space where a magnetic field is changing, there
  must an exist a looping electric field.

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Stokes Theorem and Amperes Law

This is Stokes Theorem. It holds for any field
We can use it to get Amperes Law in integral
form from Amperes Law in differential form.
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The Integral Form of Faradays Law

This says the line integral of the electric field
around an Amperian loop is minus the time
derivative of the magnetic flux through the
Amperian loop.
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Faradays Law

In other words If the number of magnetic field
lines through a loop is changing, we produce a
looping electric field.
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Example 3 A Charging Capacitor

• A capacitor with circular plates (for symmetry)
  is charging.

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A Charging Capacitor

• A normal electric field between the plates
  increases in time.

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A Charging Capacitor

• The charge increases in time but the current
  decreases in time.

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A Charging Capacitor

• On the top plate, the current is outward from the
  center. Since this current decreases, the
  acceleration is toward the center.

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A Charging Capacitor

• On the bottom plate, the current is inward toward
  the center. Since this current decreases, the
  acceleration is away from the center.

• A charge () on the bottom experience an
  acceleration toward the exit wire.

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A Charging Capacitor

• Now lets find the electric acceleration field
  from the charge on the top...

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A Charging Capacitor

• and the magnetic acceleration field from the
  charge on the top. comes out of the screen.

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A Charging Capacitor

• Now lets find the electric acceleration field
  from the charge on the bottom...

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A Charging Capacitor

• and the magnetic acceleration field from the
  charge on the bottom. comes out of the
  screen.

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A Charging Capacitor

• We can conclude that in a charging capacitor,
  magnetic field lines go around in circles in the
  capacitor, just as if real current were passing
  between the capacitor plates.

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Displacement Current

• No real charges pass between the plates of the
  capacitor, but we say that displacement current
  between the plates of the capacitor causes the
  magnetic field.

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Displacement Current

• The only real thing between the plates is an
  electric field.
• But a constant electric field cant be the
  displacement current, because there is no
  magnetic field between the capacitor plates when
  the plates are fully charged.

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Displacement Current

• We might guess that the displacement current is
  related to a changing electric field.
• Guided by Faradays Law, we might expect

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Amperes Law

• Adding this to Amperes Law as we know it, we
  expect

• The constant K can be determined either from the
  thread model or experimentally. Finally, we have

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Displacement Current

• Thus, the displacement current is a constant
  times the rate of change of the electric flux
  through an Amperian loop

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Amperes Law Revised

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Amperes Law

In other words If there is current through a
loop or the net number of electric field lines
through a loop is changing, we produce a looping
magnetic field.
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Amperes Law Revised
Warning Philosophical diversion!

• The electric field in the last term is net
  electric field of all the charges.
• A wire with a constant current produces no net
  electric field, so the last term is zero.

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Amperes Law Rewritten
Warning Philosophical diversion!

• Because the electric field is the net field, we
  can also write this as

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Amperes Law Another Viewpoint
Warning Philosophical diversion!

• A single charge moving at constant speed produces
  a displacement current, because the electric flux
  (number of field lines) from the charge passing
  through a loop changes in time.

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Amperes Law Another Viewpoint
Warning Philosophical diversion!

• A single charge moving at constant speed produces
  a displacement current, because the electric flux
  (number of field lines) from the charge passing
  through a loop changes in time.

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Amperes Law Another Viewpoint
Warning Philosophical diversion!

• We can instead calculate d/dt of the electric
  flux from each source charge and then add the
  results together.
• When we do this, the term is no longer
  needed.

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Amperes Law
Warning Philosophical diversion!

In other words If the number of electric field
lines from individual charges through a loop is
changing, we produce a looping magnetic field.
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Stokes Theorem and Amperes LawGeneralized

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Maxwells Term

• The part of Amperes Law that comes from
  displacement current is called Maxwells Term of
  Amperes Law.

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Maxwells Equations

• In the 1860s, James Clerk Maxwell added his term
  to Amperes Law and organized the known relations
  about electric and magnetic fields together in a
  mathematical form.

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Maxwells Equations in Integral Form

• Gausss Law of Electricity
• Gausss Law of Magnetism
• Amperes Law
• Faradays Law

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Maxwells Equations in Differential Form

• Gausss Law of Electricity
• Gausss Law of Magnetism
• Amperes Law
• Faradays Law

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Maxwells Equations

• Well learn how to use these new equations in
  coming chapters. For now, you simply need to see
  how accelerating charges lead to electric and
  magnetic fields with curl.